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I know it is not solvable in terms of radicals, and I don't know if it is known if it can be solved with only elementary functions of any kind. But can it be solved using only elementary functions and the W Lambert function? I know the general quintic is solvable with the generalized hypergeometric function, but I am wondering if it is possible to limit ourselves to only the W Lambert function. I have gotten close to solving $x^5+x+a=0$ using $e^x$ and the W Lambert function, so I suspect it is possible, but with my limited math knowledge it very well might not be. I am not in college yet, but want to learn about math, and regardless of if I am right or wrong, I will probably learn something. If I am right, is it known what the highest degree of polynomial that can be solved this way is?

EDIT: Probably not. They are not solvable with elementry functions. I dont know why the W function would be different

Colonizor48
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  • Could you clarify "I have gotten close to solving $x^5+x+a = 0$ using $e^x$ and the $W$ Lambert function" ? Thanks – Claude Leibovici Mar 10 '22 at 04:41
  • As in the equation has been close to having x on one side. But it is always messy. – Colonizor48 Mar 10 '22 at 19:52
  • For solutions in the elementary functions, see e.g. https://math.stackexchange.com/questions/1828551/polynomials-with-degree-5-solvable-in-elementary-functions/4225322#4225322. See also e.g. https://math.stackexchange.com/questions/617125/solving-5th-degree-or-higher-equations/671287#671287 https://math.stackexchange.com/questions/1555743/how-do-you-solve-5th-degree-polynomials/1557660#1557660 – IV_ Mar 19 '22 at 14:38
  • Lambert W inverts a transcendental equation while you have an algebraic equation of degree $5$ – Тyma Gaidash Sep 13 '22 at 22:42

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In the case $x^5 + x +a =0$ you can solve it in terms of generalization of Lambert Function, called Lambert-Tsallis function, and the operations ( +, - , *, / and ^). See some of my answers, specially this one.

ZKZ
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